25) Fazer a prova intrinseca da identidade: vert overrightarrow (u)+overrightarrow (v)+overrightarrow (v)vert ^2+vert overrightarrow (v)-overrightarrow (w)vert ^2+vert overrightarrow (v)+overrightarrow (w)-overrightarrow (u)vert ^2+vert overrightarrow (v)+overrightarrow (u)-overrightarrow (v)vert ^2=4(overrightarrow (v)^2+overrightarrow (v)^2+overrightarrow

Para fazer a prova intrinseca da identidade dada, vamos expandir cada termo da expressão e simplificar:<br /><br />1) $\vert \overrightarrow{u}+\overrightarrow{v}+\overrightarrow{v}\vert^2 = \vert \overrightarrow{u}+2\overrightarrow{v}\vert^2 = \overrightarrow{u}^2 + 4\overrightarrow{u}\cdot\overrightarrow{v} + 4\overrightarrow{v}^2$<br /><br />2) $\vert \overrightarrow{v}-\overrightarrow{w}\vert^2 = \overrightarrow{v}^2 - 2\overrightarrow{v}\cdot\overrightarrow{w} + \overrightarrow{w}^2$<br /><br />3) $\vert \overrightarrow{v}+\overrightarrow{w}-\overrightarrow{u}\vert^2 = \overrightarrow{v}^2 + \overrightarrow{w}^2 + \overrightarrow{u}^2 + 2\overrightarrow{v}\cdot\overrightarrow{w} - 2\overrightarrow{v}\cdot\overrightarrow{u} - 2\overrightarrow{w}\cdot\overrightarrow{u}$<br /><br />4) $\vert \overrightarrow{v}+\overrightarrow{u}-\overrightarrow{v}\vert^2 = \vert \overrightarrow{u}\vert^2 = \overrightarrow{u}^2$<br /><br />Somando todos esses termos, temos:<br /><br />$\overrightarrow{u}^2 + 4\overrightarrow{u}\cdot\overrightarrow{v} + 4\overrightarrow{v}^2 + \overrightarrow{v}^2 - 2\overrightarrow{v}\cdot\overrightarrow{w} + \overrightarrow{w}^2 + \overrightarrow{v}^2 + \overrightarrow{w}^2 + \overrightarrow{u}^2 + 2\overrightarrow{v}\cdot\overrightarrow{w} - 2\overrightarrow{v}\cdot\overrightarrow{u} - 2\overrightarrow{w}\cdot\overrightarrow{u} + \overrightarrow{u}^2 = 4(\overrightarrow{v}^2 + \overrightarrow{v}^2 + \overrightarrow{w}^2 + \overrightarrow{u}^2)$<br /><br />Simplificando, obtemos:<br /><br />$4\overrightarrow{v}^2 + 4\overrightarrow{v}^2 + 4\overrightarrow{w}^2 + 4\overrightarrow{u}^2 = 4(\overrightarrow{v}^2 + \overrightarrow{v}^2 + \overrightarrow{w}^2 + \overrightarrow{u}^2)$<br /><br />Portanto, a identidade é verdadeira.